Cauchy Principal Value

Date: 05/23/2000 at 05:03:43
From: Derek Waldron
Subject: Cauchy Principal Value

Hi Dr. Math,

Can you tell me exactly what the Cauchy Principal Value is? I know 
it's used when a singularity lies on the contour, but all the 
literature I've read doesn't actually DEFINE what the value is.

Thanks,
Derek Waldron
Undergraduate Physics Student

Date: 05/23/2000 at 10:07:40
From: Doctor Rob
Subject: Re: Cauchy Principle Value

Thanks for writing to Ask Dr. Math, Derek.

The Cauchy Principal Value of a divergent integral is the limit as the 
radius goes to zero of the integral of the function outside a circular 
neighborhood of the singularity.

The simple examples are Riemann integrals along the real line. For 
example, the integral of y = x + 1/x from -1 to 2 diverges because 
there is a singularity at x = 0. Then the CPV of the integral is the 
limit of the integral outside the interval (-r,r), as r -> 0:

                        -r                     2
     CPV = lim  INTEGRAL  (x+1/x) dx + INTEGRAL (x+1/x) dx
           r->0         -1                     r

                              -r                    2
         = lim [x^2 + ln(|x|)]     + [x^2 + ln(|x|)]   
           r->0               x=-1                  x=r

         = lim r^2 + ln(r) - 1 - 0 + 4 + ln(2) - r^2 - ln(r)
           r->0

         = lim  3 + ln(2)
           r->0

         = 3 + ln(2)

If we took, instead, the interval (-r,2*r), we would get a different 
limit. This is an artifact of the divergence of the integral.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/